A wavelength coincidence that weighs a neutron star

A shift in a line is a speedometer that works at any distance, and in a binary star it is a scale. Two stars orbiting each other move in opposite directions along the line of sight, their spectral lines shift back and forth with the orbit, and the amplitudes of those shifts, with the period, give the stars’ masses up to the unknown inclination of the orbit. That is how the only stars whose masses are known were weighed, and how the first black holes were identified: a star seen orbiting an unseen companion too fast for the companion to be anything but collapsed.

The method needs a star that can be seen. In the brightest X-ray binaries — systems where a neutron star or black hole pulls gas from a low-mass companion through a disc that glows at millions of degrees — the companion cannot be seen. The disc outshines it in visible light by a factor of a hundred or more, and the absorption lines that would give its velocity are buried. The compact star is invisible by nature; its companion is invisible by circumstance; and a binary with neither star visible cannot be weighed.

A coincidence of atomic physics, noticed in the 1930s for an unrelated reason, gets round this. It makes the companion’s surface emit lines of its own — not from its own light, but from the X-rays falling on it — and those lines move with the companion.

Two lines a sixtieth of an ångström apart #

Ionised helium, He⁺, has the hydrogen-like spectrum of a single electron round a doubly charged nucleus, and its version of the Lyman-α line lies in the far ultraviolet at 303.783 Å. Twice-ionised oxygen, O²⁺, has a resonance line at 303.799 Å. The two wavelengths differ by sixteen thousandths of an ångström, which in velocity is sixteen kilometres a second.

In gas hot enough to ionise helium twice, the He II line photons are produced in great numbers whenever He²⁺ recombines, and in a dense enough gas they are scattered back and forth many times before escaping. Thermal motion broadens the helium line by about ten kilometres a second at twenty thousand kelvin and twenty at a hundred thousand — more than the oxygen line, because helium is four times lighter. The two profiles overlap, and on every scattering there is a chance that the photon is absorbed instead by O²⁺. The overlap is only partial, but because the helium photon is scattered so many times, a large fraction of all He II Lyman-α photons ends up pumping oxygen.

Ira Bowen worked this out in 1934 to explain why certain O III lines in planetary nebulae were far stronger than collisions or recombination could make them. An oxygen ion excited by a He II photon decays in a cascade through a sequence of levels, emitting lines in the near ultraviolet and the blue, and one of those cascades ends with a photon at 374 Å that happens to coincide with a line of N III — a second coincidence — so nitrogen is pumped in turn and emits its own cascade, including lines near 4634 and 4641 Å in the blue. The Bowen blend near 4640 Å, with the C III lines that sit beside it, is the visible signature of the whole chain.

A glow from the face that sees the X-rays #

In an X-ray binary — a companion that has filled the surface it can no longer hold and pours gas through a disc that has to throw its angular momentum away — the X-rays from the inner disc fall on everything nearby, including the companion star’s surface on the side facing the compact object. That face is heated and ionised: its outer layers are a hot plasma bathed in X-rays, full of He²⁺, O²⁺ and N²⁺ — exactly the conditions for Bowen fluorescence. The lit face therefore emits the 4640 Å blend, in narrow lines, because the gas on the stellar surface moves with the star rather than whirling round the compact object as the disc’s gas does.

The disc emits the blend too, but its lines are broad, smeared over hundreds of kilometres a second by the disc’s rotation, and its continuum is a stack of temperatures that swamps any absorption line the companion could show. The narrow components, a few tens of kilometres a second wide, sit on top of the broad ones and move back and forth with the orbital period. They were first separated in 2002 in the spectrum of Sco X-1, the brightest X-ray source in the sky after the Sun, by taking many spectra round the orbit and combining them in the companion’s frame. The donor star in Sco X-1 had never been seen. Its Bowen lines moved with an amplitude of about seventy-seven kilometres a second.

Lines that follow the star but not exactly #

The narrow lines move with the companion but they are not emitted from its centre.

The lit face lies on the side of the companion nearer the compact object, which is closer to the binary’s centre of mass than the companion’s own centre is. Points closer to the centre of mass move more slowly round it. So the lit face’s velocity amplitude, $K_\mathrm{em}$, is smaller than the companion’s centre-of-mass amplitude, $K_2$, and the amplitude measured from the narrow lines is a lower limit on the one the masses need. For the model system in the figure the lit face moves at a hundred and nine kilometres a second against the companion’s hundred and thirty-nine — a difference of a fifth, which becomes a difference of nearly a factor of two in the mass that follows, because the mass goes as the cube of the velocity.

The broad lines of the disc, by contrast, are supposed to trace the compact star, moving in antiphase. They do so only roughly: the disc is not symmetric, the stream of gas from the companion hits it at one side, and the broad lines’ velocities are distorted by that asymmetry by tens of kilometres a second. The narrow lines from the companion are the cleaner clock, provided the offset of the lit face can be corrected for — lines that disagree about a velocity are a problem everywhere spectroscopy weighs stars, and here the disagreement has a geometry.

How much slower the lit face moves #

The size of the correction depends on how large the companion is relative to the orbit and on which part of its face is lit.

The companion fills its Roche lobe — that is why it is losing mass — and the lobe’s size relative to the separation depends only on the mass ratio. A more massive companion, relative to the compact object, has a larger lobe, its lit face is further from its centre, and the correction is larger. Where on the face the light comes from depends on how the X-rays reach it. The disc, which is thicker at its outer edge, casts a shadow across the companion’s equator, so the brightest Bowen emission comes from the parts of the face above and below the shadow, further from the inner Lagrange point than an unshadowed face would emit. Models of the irradiated face with a disc of different thicknesses give the correction as a function of the mass ratio and the disc’s opening angle, and the range of those models is the main uncertainty in the method.

A neutron star weighed through its companion’s face #

With a correction in hand, the companion’s velocity amplitude and the period give the mass function, and with the mass ratio and the inclination, the compact star’s mass.

Three uncertainties stack. The uncorrected amplitude gives a hard lower limit on the mass. The correction raises it by tens of per cent to nearly a factor of three, depending on the shadow and the mass ratio. And the inclination enters as the cube of its sine, so that between thirty and fifty degrees the implied mass changes by a factor of more than three. In Sco X-1 the inclination is constrained by the radio jets, whose orientation on the sky and relative brightness give an angle of about forty-four degrees to the line of sight, and the result is consistent with a neutron star of ordinary mass — near the 1.4 solar masses set by a star held up by a counting rule — which, for a system weighed through a lit face whose star has never been seen, is a confirmation of the method rather than a surprise.

The technique has since been applied to dozens of X-ray binaries in outburst, when their discs are brightest and the irradiation strongest. In the black-hole candidate GX 339-4, whose companion had never been detected, Bowen lines gave a mass function of about six solar masses — a lower limit on the compact object’s mass above the maximum mass of a neutron star — establishing it as a black hole from a measurement of light that the companion does not itself produce. It joined the short list of stellar black holes with dynamical masses, a list that is itself selected, since the black holes that can be weighed are the heavy ones.

Where the angle is known #

The inclination is the largest uncertainty in the mass, and eclipsing systems remove it. When the companion passes in front of the disc once an orbit, the orbit must be nearly edge-on, and the depth and shape of the eclipse fix the angle to a few degrees.

A shorter orbit and a nearly edge-on view make every velocity larger, and the lit face’s lines sweep over more than five hundred kilometres a second each orbit. In such a system the Bowen lines can be measured with high precision even in a few nights, and the shape of the irradiated face can be checked against the eclipse light curve, which shows directly how the lit face and the disc’s rim are arranged.

Once the inclination is fixed near eighty degrees, its sine cubed changes by only a few per cent across the allowed range, and almost all the remaining uncertainty is in the correction for the lit face’s offset. At the eclipsing angle the three curves give 0.74, 1.16 and 1.97 solar masses: the eclipse has removed the inclination, and what is left is a factor of more than two from the correction alone. That is the sense in which the method is limited by the shadow rather than by the spectroscopy. The amplitude itself can be measured to a few kilometres a second; turning it into a mass needs to know where on the companion’s face the light comes from, and the answer depends on how thick the outer disc is. In X 1822-371 the eclipse light curve itself shows a thick disc rim, which pushes the emission away from the inner Lagrange point and towards the larger corrections, and the published analysis, with the shadow modelled from the light curve, put the neutron star between about 1.6 and 2.3 solar masses — a case where the geometry, not the velocity, set the width of the answer.

Eclipsing systems are rare, a few per cent of the whole, because only a narrow range of angles gives an eclipse. They serve as calibrators. A correction tested in an eclipsing system, where the geometry is known, can be applied to the much more numerous systems seen at intermediate angles, where it cannot be tested directly.

Why the outburst is the time to look #

The method has a timing that is the reverse of the usual one. Most measurements of companion stars in X-ray binaries are made in quiescence, when the disc has faded and the companion’s own absorption lines can be seen. In quiescence the X-rays are weak, the companion’s face is not lit, and there is no Bowen fluorescence. In outburst the companion’s own lines are swamped, and the Bowen lines are bright. The two methods are complementary in time: one works when the system is dim, the other when it is bright. Many X-ray binaries — Sco X-1 among them — are persistently bright and never enter quiescence, and for those the Bowen lines are the only route to the companion’s motion that exists.

It also means that the measurement is made on a star whose outer layers are far from normal. The lit face’s temperature is set by the X-ray flux rather than by the star’s own interior, and it can be tens of thousands of degrees on a star whose own surface is a few thousand. The line emission is a reprocessing of the X-rays, and its strength rises and falls with the X-ray luminosity. Variations in the narrow lines’ strength round the orbit, as the lit face turns towards and away from the observer, give the inclination independently in some systems — a lit hemisphere seen face-on near the companion’s superior conjunction and edge-on at quadrature — which can break the degeneracy between the mass and the inclination that the velocities alone leave.

Two coincidences, and why astronomy needs them #

The Bowen mechanism rests on two near-coincidences of wavelength in three different ions, neither of which has any reason to be there. The He II Lyman-α wavelength is set by the hydrogen-like energy levels of helium; the O III line by the complicated electronic structure of a six-electron ion; the N III line by that of a five-electron one. That they fall within a few hundredths of an ångström of each other is an accident of atomic structure, and fluorescence by line coincidence is otherwise rare in astrophysics. The few cases that matter — Bowen’s, and the pumping of Fe II by hydrogen Lyman-α in some stars and nebulae — are exploited precisely because they turn a spectrum the observer cannot see into one that can be seen.

The same is true, in a broader sense, of most of the tools that weigh compact objects. A neutron star in an X-ray binary is weighed by its companion; the companion, here, by the X-rays the neutron star produces; and the X-rays, through atomic physics, by the visible lines they excite. Each step is a measurement of one thing made with the light of another, and the only light the observer receives from the system’s mass is the light the system’s own energy makes.

One temperature, one sinusoid and one number for the shadow #

The figures use idealised versions of each piece. The line profiles are pure Gaussians with a single temperature and turbulence, while real Bowen emission comes from layers with a range of temperatures and densities, and its efficiency depends on the optical depth in He II Lyman-α, which sets how many times each photon is scattered before it escapes or is absorbed. The radial velocity curves are sinusoids for a circular orbit with no contribution from the disc’s own emission, and the K-correction is reduced to a single number for the offset of the lit face, where real models integrate the irradiated, partially shadowed, Roche-lobe-shaped surface. And the mass figure holds the mass ratio fixed, while in a real system it is inferred from the rotational broadening of the narrow lines or from other constraints, with its own uncertainty. All of these are modelled in the published analyses; none of them is small.

Still open: how much of the face the disc hides #

The K-correction is the weak link, and it is set by a part of the system nobody has resolved: the vertical structure of the outer accretion disc, whose thickness decides how much of the companion’s face is shadowed. A thick disc shadows the equator and moves the emission to higher latitudes; a thin one lets the whole face be lit. Eclipsing systems, where the inclination is known and the companion’s shape can be modelled, are the calibration: in them the corrected Bowen amplitude can be compared with the velocity from the companion’s own absorption lines in quiescence. Where both have been measured they agree within the range the corrections allow, but the range is broad. If the comparison is made for enough systems to fix the disc’s opening angle as a function of the X-ray luminosity, the Bowen method becomes a precision tool for neutron-star masses in exactly the bright systems no other method reaches. If not, it remains what it has been since 2002 — a way to prove that an invisible star exists and is moving, and to set a floor under the mass of the thing it orbits.